Suppose $k$ is a fixed commutative ring, and let $\mathsf{dgCat}_k$ denote the category of $k$-linear dg-categories. We will equip $\mathsf{dgCat}_k$ with the Morita model structure (see theorem 2.27 here or theorem 2.22 here). The category $\mathsf{dgCat}_k$ admits a symmetric monoidal structure, as defined in section 4 here.
$\mathsf{dgCat}_k$ equipped with this symmetric monoidal structure is not a monoidal model category. As such, we cannot apply results of Schwede-Shipley or White to produce a model category structure on algebra objects in $\mathsf{dgCat}_k.$
However, a result of Cohn says that the underlying $\infty$-category of $\mathsf{dgCat}_k$ localized at the Morita equivalences is equivalent to the $\infty$-category of small idempotent-complete $k$-linear stable $\infty$-categories. The latter category does (I believe) have a symmetric monoidal structure induced by the Lurie tensor product of stable $\infty$-categories (this is claimed, for example, in section 10.4 here). This implies the existence of a sensible category of commutative algebras in the $\infty$-category of dg-categories.
My question is, how can we talk about such things at the level of model categories? I would like to be able to talk about the homotopy pushout of a diagram of commutative (or maybe $E_\infty$) algebras in $\mathsf{dgCat}_k$, but the standard results do not guarantee that the category of such objects would admit a model structure. (I want to construct a relative derived tensor product of commutative algebras in dg-categories, and I want the result to retain the algebra structure, instead of only having a module structure.) Given the equivalence of $\infty$-categories above, it seems that there may be some reasonable way to talk about such things at the model-category level. Or am I asking for too much?
More generally, is there any "standard" way of discussing derived/homotopy coherent notions of algebras in a symmetric monoidal category which is also a model category (but which is not a monoidal model category)?
